Question

question 4: solve the exponential equation.
$125^{x - 2}=5^{x}$
$x = \frac{1}{2}$ option 1
$x = \frac{1}{3}$ option 2
$x = 2$ option 3
$x = 3$ option 4

Explanation:

Step1: Express 125 as a power of 5

Since \(125 = 5^3\), we can rewrite the left - hand side of the equation \(125^{x - 2}=5^{x}\) as \((5^{3})^{x - 2}\).
Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), we have \((5^{3})^{x - 2}=5^{3(x - 2)}\). So the equation becomes \(5^{3(x - 2)}=5^{x}\).

Step2: Set the exponents equal

For the exponential equation \(a^{m}=a^{n}\) (where \(a>0,a
eq1\)), we know that \(m = n\).
Since \(a = 5\) (which satisfies \(5>0\) and \(5
eq1\)) in our equation \(5^{3(x - 2)}=5^{x}\), we can set the exponents equal: \(3(x - 2)=x\).

Step3: Solve the linear equation

Expand the left - hand side: \(3x-6 = x\).
Subtract \(x\) from both sides: \(3x - x-6=x - x\), which simplifies to \(2x-6 = 0\).
Add 6 to both sides: \(2x-6 + 6=0 + 6\), so \(2x=6\)? Wait, no, wait. Wait, let's go back. Wait, \(3(x - 2)=x\) expands to \(3x-6=x\). Then subtract \(x\) from both sides: \(3x - x-6=x - x\) gives \(2x-6 = 0\)? No, wait, \(3x-6=x\), then \(3x-x=6\), \(2x = 6\)? Wait, no, that's a mistake. Wait, no, let's re - do step 2 and 3.

Wait, starting over from step 2:

We have \(5^{3(x - 2)}=5^{x}\), so \(3(x - 2)=x\).

Expand \(3(x - 2)\): \(3x-6=x\).

Subtract \(x\) from both sides: \(3x - x-6=x - x\) \(\Rightarrow2x-6 = 0\)? No, \(3x - x=6\) \(\Rightarrow2x=6\)? Wait, no, that would give \(x = 3\), but that's not correct. Wait, I made a mistake. Wait, \(125^{x - 2}=5^{x}\), \(125 = 5^3\), so \((5^3)^{x - 2}=5^{3(x - 2)}\), so \(5^{3x-6}=5^{x}\). Then \(3x - 6=x\). Subtract \(x\) from both sides: \(2x-6 = 0\), add 6 to both sides: \(2x=6\), \(x = 3\)? Wait, no, that's not matching the options. Wait, no, wait the original equation is \(125^{x - 2}=5^{x}\). Wait, maybe I misread the equation. Wait, is it \(125^{x}-2 = 5^{x}\) or \(125^{x - 2}=5^{x}\)? The user wrote \(125^{x-2}=5^{x}\). Wait, let's check the options. The options are \(x=\frac{1}{2}\), \(x = \frac{1}{3}\), \(x = 2\), \(x = 3\).

Wait, maybe I made a mistake in the base. Wait, \(125=5^3\), so \(125^{x - 2}=(5^3)^{x - 2}=5^{3(x - 2)}\). So \(5^{3(x - 2)}=5^{x}\) implies \(3(x - 2)=x\).

\(3x-6=x\)

\(3x - x=6\)

\(2x=6\)

\(x = 3\). But let's check with \(x = 3\): left - hand side \(125^{3 - 2}=125^{1}=125\), right - hand side \(5^{3}=125\). So \(x = 3\) is correct. Wait, but the options have \(x = 3\) as Option 4.

Wait, maybe the original equation was \(125^{x}-2 = 5^{x}\)? No, the user wrote \(125^{x-2}=5^{x}\). Let's check again.

Wait, if the equation is \(125^{x-2}=5^{x}\), then:

\(5^{3(x - 2)}=5^{x}\)

\(3x-6=x\)

\(2x=6\)

\(x = 3\). So Option 4: \(x = 3\) is correct.

Answer:

Option 4: \(x = 3\)